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Type | Label | Description |
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Statement | ||
Theorem | vtxnbuvtx 27101* | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) | ||
Theorem | uvtxnbgrss 27102 | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁)) | ||
Theorem | uvtxnbgrvtx 27103* | A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣)) | ||
Theorem | uvtx0 27104 | There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅) | ||
Theorem | isuvtx 27105* | The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒} | ||
Theorem | uvtxel1 27106* | Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) | ||
Theorem | uvtx01vtx 27107 | If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1)) | ||
Theorem | uvtx2vtx1edg 27108* | If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 25-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((♯‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | ||
Theorem | uvtx2vtx1edgb 27109* | If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) | ||
Theorem | uvtxnbgr 27110 | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) | ||
Theorem | uvtxnbgrb 27111 | A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))) | ||
Theorem | uvtxusgr 27112* | The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ 𝐸}) | ||
Theorem | uvtxusgrel 27113* | A universal vertex, i.e. an element of the set of all universal vertices, of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ 𝐸))) | ||
Theorem | uvtxnm1nbgr 27114 | A universal vertex has 𝑛 − 1 neighbors in a finite graph with 𝑛 vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘(𝐺 NeighbVtx 𝑁)) = ((♯‘𝑉) − 1)) | ||
Theorem | nbusgrvtxm1uvtx 27115 | If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑈 ∈ (UnivVtx‘𝐺))) | ||
Theorem | uvtxnbvtxm1 27116 | A universal vertex has 𝑛 − 1 neighbors in a finite simple graph with 𝑛 vertices. A biconditional version of nbusgrvtxm1uvtx 27115 resp. uvtxnm1nbgr 27114. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (𝑈 ∈ (UnivVtx‘𝐺) ↔ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1))) | ||
Theorem | nbupgruvtxres 27117* | The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))) | ||
Theorem | uvtxupgrres 27118* | A universal vertex is universal in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝐾 ∈ (UnivVtx‘𝐺) → 𝐾 ∈ (UnivVtx‘𝑆))) | ||
Syntax | ccplgr 27119 | Extend class notation with (arbitrary) complete graphs. |
class ComplGraph | ||
Syntax | ccusgr 27120 | Extend class notation with complete simple graphs. |
class ComplUSGraph | ||
Definition | df-cplgr 27121 | Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e., each vertex has all other vertices as neighbors or, in other words, each vertex is a universal vertex. (Contributed by AV, 24-Oct-2020.) (Revised by TA, 15-Feb-2022.) |
⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} | ||
Definition | df-cusgr 27122 | Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 27164. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by BJ, 14-Feb-2022.) |
⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | ||
Theorem | cplgruvtxb 27123 | A graph 𝐺 is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 15-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) | ||
Theorem | prcliscplgr 27124* | A proper class (representing a null graph, see vtxvalprc 26758) has the property of a complete graph (see also cplgr0v 27137), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 27125. (Contributed by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | ||
Theorem | iscplgr 27125* | The property of being a complete graph. (Contributed by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) | ||
Theorem | iscplgrnb 27126* | A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) | ||
Theorem | iscplgredg 27127* | A graph 𝐺 is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑣, 𝑛} ⊆ 𝑒)) | ||
Theorem | iscusgr 27128 | The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | ||
Theorem | cusgrusgr 27129 | A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | ||
Theorem | cusgrcplgr 27130 | A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.) |
⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) | ||
Theorem | iscusgrvtx 27131* | A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) | ||
Theorem | cusgruvtxb 27132 | A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉)) | ||
Theorem | iscusgredg 27133* | A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ 𝐸)) | ||
Theorem | cusgredg 27134* | In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | ||
Theorem | cplgr0 27135 | The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.) |
⊢ ∅ ∈ ComplGraph | ||
Theorem | cusgr0 27136 | The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
⊢ ∅ ∈ ComplUSGraph | ||
Theorem | cplgr0v 27137 | A null graph (with no vertices) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph) | ||
Theorem | cusgr0v 27138 | A graph with no vertices and no edges is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph) | ||
Theorem | cplgr1vlem 27139 | Lemma for cplgr1v 27140 and cusgr1v 27141. (Contributed by AV, 23-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((♯‘𝑉) = 1 → 𝐺 ∈ V) | ||
Theorem | cplgr1v 27140 | A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((♯‘𝑉) = 1 → 𝐺 ∈ ComplGraph) | ||
Theorem | cusgr1v 27141 | A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((♯‘𝑉) = 1 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph) | ||
Theorem | cplgr2v 27142 | An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸)) | ||
Theorem | cplgr2vpr 27143 | An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 3-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝐺 ∈ UHGraph ∧ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ ComplGraph ↔ {𝐴, 𝐵} ∈ 𝐸)) | ||
Theorem | nbcplgr 27144 | In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) | ||
Theorem | cplgr3v 27145 | A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))) | ||
Theorem | cusgr3vnbpr 27146* | The neighbors of a vertex in a simple graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 5-Nov-2020.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ USGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧})) | ||
Theorem | cplgrop 27147 | A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.) |
⊢ (𝐺 ∈ ComplGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ ComplGraph) | ||
Theorem | cusgrop 27148 | A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.) |
⊢ (𝐺 ∈ ComplUSGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ ComplUSGraph) | ||
Theorem | cusgrexilem1 27149* | Lemma 1 for cusgrexi 27153. (Contributed by Alexander van der Vekens, 12-Jan-2018.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V) | ||
Theorem | usgrexilem 27150* | Lemma for usgrexi 27151. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | ||
Theorem | usgrexi 27151* | An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 10-Nov-2021.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph) | ||
Theorem | cusgrexilem2 27152* | Lemma 2 for cusgrexi 27153. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒) | ||
Theorem | cusgrexi 27153* | An arbitrary set 𝑉 regarded as set of vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ ComplUSGraph) | ||
Theorem | cusgrexg 27154* | For each set there is a set of edges so that the set together with these edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
⊢ (𝑉 ∈ 𝑊 → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) | ||
Theorem | structtousgr 27155* | Any (extensible) structure with a base set can be made a simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (♯‘𝑥) = 2} & ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ 𝐺 = (𝑆 sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → 𝐺 ∈ USGraph) | ||
Theorem | structtocusgr 27156* | Any (extensible) structure with a base set can be made a complete simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) (Proof shortened by AV, 14-Feb-2022.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (♯‘𝑥) = 2} & ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ 𝐺 = (𝑆 sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → 𝐺 ∈ ComplUSGraph) | ||
Theorem | cffldtocusgr 27157* | The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} & ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) ⇒ ⊢ 𝐺 ∈ ComplUSGraph | ||
Theorem | cusgrres 27158* | Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph) | ||
Theorem | cusgrsizeindb0 27159 | Base case of the induction in cusgrsize 27164. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = ((♯‘𝑉)C2)) | ||
Theorem | cusgrsizeindb1 27160 | Base case of the induction in cusgrsize 27164. The size of a (complete) simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = ((♯‘𝑉)C2)) | ||
Theorem | cusgrsizeindslem 27161* | Lemma for cusgrsizeinds 27162. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1)) | ||
Theorem | cusgrsizeinds 27162* | Part 1 of induction step in cusgrsize 27164. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹))) | ||
Theorem | cusgrsize2inds 27163* | Induction step in cusgrsize 27164. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⇒ ⊢ (𝑌 ∈ ℕ0 → ((𝐺 ∈ ComplUSGraph ∧ (♯‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉) → ((♯‘𝐹) = ((♯‘(𝑉 ∖ {𝑁}))C2) → (♯‘𝐸) = ((♯‘𝑉)C2)))) | ||
Theorem | cusgrsize 27164 | The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2)) | ||
Theorem | cusgrfilem1 27165* | Lemma 1 for cusgrfi 27168. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺)) | ||
Theorem | cusgrfilem2 27166* | Lemma 2 for cusgrfi 27168. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} & ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) ⇒ ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃) | ||
Theorem | cusgrfilem3 27167* | Lemma 3 for cusgrfi 27168. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} & ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin)) | ||
Theorem | cusgrfi 27168 | If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin) | ||
Theorem | usgredgsscusgredg 27169 | A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐻) & ⊢ 𝐹 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → 𝐸 ⊆ 𝐹) | ||
Theorem | usgrsscusgr 27170* | A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐻) & ⊢ 𝐹 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∀𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 𝑒 = 𝑓) | ||
Theorem | sizusglecusglem1 27171 | Lemma 1 for sizusglecusg 27173. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐻) & ⊢ 𝐹 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸–1-1→𝐹) | ||
Theorem | sizusglecusglem2 27172 | Lemma 2 for sizusglecusg 27173. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐻) & ⊢ 𝐹 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin) | ||
Theorem | sizusglecusg 27173 | The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐻) & ⊢ 𝐹 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹)) | ||
Theorem | fusgrmaxsize 27174 | The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2)) | ||
Syntax | cvtxdg 27175 | Extend class notation with the vertex degree function. |
class VtxDeg | ||
Definition | df-vtxdg 27176* | Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition +𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.) |
⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))) | ||
Theorem | vtxdgfval 27177* | The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 ⇒ ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))) | ||
Theorem | vtxdgval 27178* | The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 ⇒ ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) | ||
Theorem | vtxdgfival 27179* | The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 ⇒ ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) | ||
Theorem | vtxdgop 27180 | The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.) |
⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) | ||
Theorem | vtxdgf 27181 | The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*) | ||
Theorem | vtxdgelxnn0 27182 | The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑋) ∈ ℕ0*) | ||
Theorem | vtxdg0v 27183 | The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) | ||
Theorem | vtxdg0e 27184 | The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph (see also the induction steps vdegp1ai 27246, vdegp1bi 27247 and vdegp1ci 27248). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) | ||
Theorem | vtxdgfisnn0 27185 | The degree of a vertex in a graph of finite size is a nonnegative integer. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 ⇒ ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0) | ||
Theorem | vtxdgfisf 27186 | The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ0) | ||
Theorem | vtxdeqd 27187 | Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.) |
⊢ (𝜑 → 𝐺 ∈ 𝑋) & ⊢ (𝜑 → 𝐻 ∈ 𝑌) & ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) & ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) ⇒ ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) | ||
Theorem | vtxduhgr0e 27188 | The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 27184. (Contributed by AV, 15-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) | ||
Theorem | vtxdlfuhgr1v 27189* | The degree of the vertex in a loop-free hypergraph with one vertex is 0. (Contributed by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0)) | ||
Theorem | vdumgr0 27190 | A vertex in a multigraph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉 ∧ (♯‘𝑉) = 1) → ((VtxDeg‘𝐺)‘𝑁) = 0) | ||
Theorem | vtxdun 27191 | The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 19-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → Fun 𝐽) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁))) | ||
Theorem | vtxdfiun 27192 | The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → Fun 𝐽) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) & ⊢ (𝜑 → dom 𝐼 ∈ Fin) & ⊢ (𝜑 → dom 𝐽 ∈ Fin) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁))) | ||
Theorem | vtxduhgrun 27193 | The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) & ⊢ (𝜑 → 𝐻 ∈ UHGraph) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁))) | ||
Theorem | vtxduhgrfiun 27194 | The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) & ⊢ (𝜑 → 𝐻 ∈ UHGraph) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽)) & ⊢ (𝜑 → dom 𝐼 ∈ Fin) & ⊢ (𝜑 → dom 𝐽 ∈ Fin) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁))) | ||
Theorem | vtxdlfgrval 27195* | The value of the vertex degree function for a loop-free graph 𝐺. (Contributed by AV, 23-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) | ||
Theorem | vtxdumgrval 27196* | The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) | ||
Theorem | vtxdusgrval 27197* | The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = dom 𝐼 & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) | ||
Theorem | vtxd0nedgb 27198* | A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) | ||
Theorem | vtxdushgrfvedglem 27199* | Lemma for vtxdushgrfvedg 27200 and vtxdusgrfvedg 27201. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒})) | ||
Theorem | vtxdushgrfvedg 27200* | The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) +𝑒 (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))) |
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